A Colorful Journey through Endless Patterns of Quick Wits
Home  /  History  /  Two Oval Stools to a Table

Two Oval Stools to a Table*

Dissection Puzzles

by Serhiy Grabarchuk

Centuries of the Ovals
This well-known, classic puzzle with transformation of two oval stools into a circular table top has long and interesting history. More than 180 years ago, in 1821, John Jackson posed in his book Rational Amusement for Winter Evenings a puzzle how to transform a circle into two hollow ovals as shown below, and proposed an 8-piece solution. Could you discover how it can be done?

Jackson’s Table & Oval Stools puzzle.
Then, at the beginning of the 20th century this puzzle attracted attention of Sam Loyd, who was the greatest America's puzzle creator. In Loyd's legendary Cyclopedia of 5000 Puzzles, Tricks & Conundrums, the puzzle was posed under the name "An Old Saw with New Teeth" as a challenge how to dissect two ovals with an oval hand hole in each of them into the fewest number of parts which can form a circular table top.

Loyd's "An Old Saw with New Teeth" puzzle as it appeared in his Cyclopedia of 5000 Puzzles, Tricks & Conundrums.
Loyd published it as a contest, showing then a better, 6-piece solution which was based on a famous Great Chinese Monad pattern. Try to find that 6-piece solution using as a hint that pattern and patterns depicted on two ovals and on the resulting circle as shown in in the illustration below. Keep in mind that diagrams shown in the illustration contain all necessary lines to make your cuts, but not every of these lines you will need to use, though.

Pattern for 6-piece solution based on the Great Chinese Monad.
In 1997 Greg Frederickson, the World's biggest expert in dissection puzzles, published an outstanding book on history and achievements in this old and comprehensive field of recreational math - Dissections: Plane & Fancy. Chapter 15 in that book is fully devoted to dissections with curved figures, and also describes the above and some other variations of the puzzle with two hollow ovals. In the book Greg shows quite different, novel 6-piece solution to Jackson's ovals.

Since the first publication of Jackson's puzzle there were numerous attempts to improve the 6-piece solution to the puzzle. Finally, in March of 2004 I was lucky to discover several new 6-piece solutions, two basic solutions containing just five(!) pieces each, more than a dozen of different modifications of these basic 5-piece solutions, and proofs that in math sense there is an infinite number of 5-piece solutions. In every of them one piece is flipped over. Finding any of 5-piece solutions is not an easy task, so try to discover one of the simplest of them. Could you do this? Hint. The diagrams below contain all necessary lines to make your cuts. Of course, not all of them are needed to be used for this. And keep in mind that in a 5-piece solution one piece is allowed to be flipped over.

Diagram for 5-piece solution.
Last but not least, there is a variation of the hollow-ovals-to-circle puzzle posed by Sam Loyd in his attempts to find solution with the least number of pieces. It also is based on the Great Chinese Monad pattern described above, and you can easily solve it keeping in mind that every oval is divided in exactly two the same pieces. The illustration below shows this Loyd's variation and all necessary lines to make you cuts. Again, please remember that you will need just some of these lines to cut both ovals.

Loyd’s variation with four pieces.


Puzzle Solutions
Notes & References
*) This article was prepared and first published specially for the 25th International Puzzle Party, Helsinki, Finland, July 22-25, 2005.
Also, this article was published in the Homage To A Pied Puzzler collection,
A K Peters, Ltd., Natick, Massachusetts, Copyright © 2009 A K Peters, Ltd. Republished with permission.
Last Updated: December 3, 2009
Posted: August 30, 2005
< Home  |  Privacy Policy  |  About Age of Puzzles  |  Contact Us  |  Link to Us
Copyright © 2005-2009 Serhiy Grabarchuk. All Rights Reserved
Design by Art of Puzzle
Serhiy Grabarchuk Puzzles